You really need to include the full question next time, I wouldn't have known what you were asking had I not already answered this very question previously.
The derivative of a function,
dxdy gives you the gradient of the
tangent at the point
x on the curve
y by definition.
So, for example: the curve
y=x2 has gradient function/derivative
dxdy=2x. That, is the gradient of the tangent at any point
x is
2x. The gradient of the tangent to the curve at
x=3 is
2(3)=6; this is the gradient of the
tangent to the curve.
Secondly:
The normal and tangent at a point on a curve are mutually perpendicular to one another.
Hence their gradients satisfy
mNmT=−1.
So, wrapping all this up:
If you are given or you found the gradient of the normal to that point, you then want to find the gradient of the tangent at that point using the perpendicularity argument. This then lets you say that
dxdy=mT=−mN1 at the point
P.
It is
incorrect to say that
dxdy=mN at P, as dy/dx does
not measure the gradient of the normal, it measures the gradient of the
tangent. So if you want to find P, you need to use the gradient of the
tangent.